| L(s) = 1 | − 2i·3-s − 9-s − 6i·11-s + 6·17-s + 2i·19-s − 4i·27-s − 12·33-s − 6·41-s − 10i·43-s − 7·49-s − 12i·51-s + 4·57-s − 6i·59-s + 14i·67-s − 2·73-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 0.333·9-s − 1.80i·11-s + 1.45·17-s + 0.458i·19-s − 0.769i·27-s − 2.08·33-s − 0.937·41-s − 1.52i·43-s − 49-s − 1.68i·51-s + 0.529·57-s − 0.781i·59-s + 1.71i·67-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593109345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593109345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.851616209863958700260998977023, −8.172631021432919553864616120885, −7.60788838758082033607254322932, −6.68281195308945160314481488920, −5.95307842161694148323001762585, −5.28142301435937642599599931765, −3.77096149761736834639654490818, −2.98866927850474552144855626409, −1.65869642098408123024634693077, −0.65002429655742763620723412684,
1.58120374473771042556784119673, 2.94778991351403219971777831753, 3.92857836024839309430758945156, 4.74599182667682973572692993289, 5.26516734667149971724317250711, 6.50593193379168764596163133492, 7.37550334194041508567538602736, 8.103774700995373890607445658339, 9.313892019799464381556970285465, 9.694416774443088210999715477646
